Some applications of the quantum Weyl groups
type: Item
instance of: scholarly article
title: Some applications of the quantum Weyl groups
zbMATH Open document ID: 0729.17009
DOI: 10.1016/0393-0440(90)90013-S
author: Yan Soibelman
author: Sergei Levendorskiĭ
published in: Journal of Geometry and Physics
publication date: 1990-01-01T00:00:00Z
review text: Let L be a simple complex Lie algebra, G a corresponding Lie group. The Weyl group \(W_ 1\) is the group generated by simple reflections in \(H^*\), H a Cartan subalgebra of L. \(W_ 1\) is isomorphic to \(W_ 2=N(T)/T\), T a maximal torus in G. The author gives a quantum analogue of \(W_ 2\) (not \(W_ 1)\), called the quantum Weyl group. It is a Hopf algebra. He applies this to obtain a formula for the universal quantum R- matrix of a simple Lie algebra. An application is given to Hecke algebras, namely that if L(\(\Lambda\)) is a q-analogue of the adjoint representation of L, and \(L(\Lambda)_ 0\subset L(\Lambda)\) is the space of zero-weight vectors, then the quantum Weyl group acts on \(L(\Lambda)_ 0\) as (generalized) Hecke algebra.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 4204613
zbMATH Keywords: quantum Weyl group
zbMATH Keywords: universal quantum R-matrix
zbMATH Keywords: Hecke algebras
reviewed by: Earl J. Taft
MaRDI profile type: Publication
cites work: Q3819193
cites work: A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation
cites work: Q3819732
cites work: Quantum deformations of certain simple modules over enveloping algebras
cites work: On quantum groups
cites work: Ribbon graphs and their invariants derived from quantum groups
cites work: Q3810318
cites work: Compact matrix pseudogroups
full work available at URL: 0393-0440(90)90013-s
OpenAlex ID: W2087971847
description: scientific article; zbMATH DE number 4204613